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Second-order logic - Wikipedia, the free encyclopedia In a second order logic that permits quantifying over functions, it is possible to write formal sentences which mean "the domain is finite" or "the domain is of countable cardinality." To say that the domain is finite, use the sentence which says that every injective function on the domain is surjective. In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. PH is the set of languages expressible by second-order logic. en.wikipedia.org /wiki/Second-order_logic   (1791 words)

PlanetMath: second order logic Second order logic refers to logics with two (or three) types where one type consists of the objects of interest and the second is either sets of those objects or functions on those objects (or both, in the three type case). This is version 5 of second order logic, born on 2002-08-28, modified 2006-03-04. Some people, chiefly Quine, have raised philisophical objections to second order logic, centering on the question of whether models require fixing some set of sets or functions as the “actual” sets or functions for the purposes of that model. planetmath.org /encyclopedia/SecondOrderLogic.html   (281 words)

Leivant. Higher Order Logic. It follows that full second order logic cannot be interpreted in weak second order logic, and that f-validity of a second order formula is reducible to truth in N of a second order formula. Quine concludes that second order logic is a mathematical theory instead of a logic. Second order validity (in the standard sense) is not second order definable, hence is not recursively enumerable. andrew.cmu.edu /~cebrown/notes/leivant.html   (4896 words)

First-order logic - Wikipedia, the free encyclopedia First-order logic (FOL) is a system of mathematical logic, extending propositional logic (equivalently, sentential logic), which is in turn extended by second-order logic. First order logic in which no atomic sentence lies in the scope of more than three quantifiers, has the same expressive power as the relation algebra of Tarski and Givant (1987). Most of these logics are in some sense extensions of first order logic: they include all the quantifiers and logical operators of first order logic with the same meanings. en.wikipedia.org /wiki/First-order_logic#An_important_type_of_well_formed_formulas:_clauses   (3132 words)

II Moore says that ``The question of which logic was appropriate for set theory--first-order logic, second-order logic, or an infinitary logic--culminated in a vigorous exchange between Zermelo and Gödel around 1930''. Second-order logic is otherwise a logic in which the set-theoretic notions, for all that has been shown, are capturable. Maybe it seems strange that a logician could be confused concerning what holds within a logic of one order and what holds within a logic of another order; but Gregory Moore (1988) actually argues forcefully that both Fraenkel and von Neumann in the 1920s were confused on the very same issue. www.hf.uio.no /filosofi/njpl/vol1no2/howlogic/node3.html   (2404 words)

Citebase - The monadic second-order logic of graphs XVI : Canonical graph<br> decompositions The modular decomposition of countable graphs:Constructions in Monadic Second-Order Logic. http://www.labri.fr/Perso/ courcell/Textes/BCOumSubmitted(2004).pdfVertex-minors, monadic second-order logic and a conjecture by Seese. The monadic second-order logic of graphs X:Linear orderings. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:cs/0510066   (469 words)

Decision Problems For Second Order Linear Logic LJ2 is second order intuitionistic propositional logic, shown undecidable by Loeb [JSL 41 (1976) 705-718] and Gabbay. Decision Problems For Second Order Linear Logic Patrick Lincoln, Andre Scedrov, and Natarajan Shankar Relating to the study of polymorphic languages based on linear logic, we have been studying fragments of second order linear logic. Subject: Decision Problems For Second Order Linear Logic www.cis.upenn.edu /~bcpierce/types/archives/1994/msg00133.html   (271 words)

Second order logic Second order logic can be given so called "weak" or "Henkin" semantics, which makes it virtually the same as first order logic, but then the quantifiers do not range any more over all subsets of the universe. Second order logic permits existential and universal quantification over subsets (and over relations and functions) of the universe. Ordinary first order predicate logic is not able take quantify over subsets of the universe even though it seems often a natural thing to do. www.math.helsinki.fi /~logic/opetus/ext_elem_logic/second_order.html   (134 words)

fomletter8.txt Logic includes the study of the form of sentences; the maneuver involved in second-order logic (quantifying over predicates) is certainly of logical interest and calls for logical analysis. Second-order logic falls within the scope of logic(1); the question of the admissibility of second-order logic as a fundamental system of reasoning is part of the analysis of predication, which is a logical(1) issue. It appears obvious that the question of the meaningfulness of second-order quantifiers falls under this definition; second-order logic is part of the subject matter of logic (though one might belong to a school of logic which regards second-order logic as inadmissible: "thou shalt not quantify over predicates"). math.boisestate.edu /~holmes/holmes/fomletter8.txt   (594 words)

--The Plan Horn existential second-order logic has the interesting property that if all graphs with n nodes are equally likely, then the probability that a random graph with n nodes will satisfy a given existential second order Horn expression approaches either 1 or 0 as n goes to infinity. By existential second order logic we refer to either expressions in first order logic with some vocabulary V, which we denote f, or expression of the form $Pf, where to f’s vocabulary we add the additional relational symbol P which takes some fixed number of arguments. Although the full proof was not given, the idea behind the characterization of NP using existential second order logic has been described. www.cs.brown.edu /courses/gs019/papers/logic.html   (3381 words)

I Certainly, such considerations have been influential when first-order logic has been preferred over second-order logic: for in virtue of the fact that first-order languages equipped with the standard semantics, as opposed to second-order languages equipped with the standard semantics, are compact, complete and have the Löwenheim-Skolem property, they have a much nicer model theory. As concepts such as finitude, well-ordering, well-foundedness and so on--which are presupposed as well understood in mathematics--can be characterized within second-order logic but not within first-order logic, second-order logic is favoured by considerations of this kind. That mathematical logicians have tended to adhere to first-order logic is for example seen from the fact that ZF and PA are almost always formalized as first-order theories. www.hf.uio.no /filosofi/njpl/vol1no2/howlogic/node2.html   (697 words)

Re: Second order logic and recursion This can be done in second- and higher-order logic by defining a predicate in the way Dedekind did for the natural numbers as the predicate which applies to those objects which satisfy all predicates which apply to zero and are closed under successor. >For the natural deduction presentation of second order logic (also for >sequent calculus) proofs of cut elimination (by Girard, Martin-Lof) >can be found in [2]. With this approach, the induction "axiom" becomes a theorem, and the logic can be used to prove that definition by recursion works (using, e.g., Kalmar's proof as presented in Landau's _Grundlagen der Analysis_). www.seas.upenn.edu /~sweirich/types/archive/1999-2003/msg00171.html   (324 words)

Frege's Logic, Theorem, and Foundations for Arithmetic Thus, Frege's second-order logic and theory of extensions together required the impossible situation in which the domain of concepts has to be strictly larger than the domain of extensions while at the same time the domain of extensions has to be as large as the domain of concepts. His axioms included familiar axioms of propositional logic, second-order predicate logic, and the logic of identity. A properly reformulated theory of ‘logical’ objects should have: (1) a separate non-logical comprehension principle which explicitly asserts the existence of logical objects, and (2) a separate identity principle which asserts the conditions under which logical objects are identical. plato.stanford.edu /entries/frege-logic   (15076 words)

fomletter13.txt Second order logic is not defined by any complete set of rules of inference. I have noted that there are more general definitions of what "logic" is that do support the idea that second-order logic is a logic. Moreover, the set of theorems of second order arithmetic whose interpretations are provable in ZFC is quite large; it is, for example, larger than the set of theorems provable in the first order two-sorted theory usually called "second-order arithmetic". math.boisestate.edu /~holmes/holmes/fomletter13.txt   (2094 words)

MONA: Monadic Second-Order Logic in Practice The purpose of this article is to introduce Monadic Second-order Logic as a practical means of specifying regularity. The logic is a highly succinct alternative to the use of regular expressions. We have built a tool Mona, which acts as a decision procedure and as a translator to finite-state machines. www.brics.dk /RS/95/21   (134 words)

Oxford University Press His coverage of the wide range of logical and philosophical topics required for understanding the controversy over second-order logic is thorough, clear, and persuasive. He then demonstrates that second-order notions are prevalent in mathematics as practised, and that higher-order logic is needed to codify many contemporary mathematical concepts. To support this contention, he first gives a detailed development of second-order and higher-order logic, in a way that will be accessible to graduate students. www.oup.com /ca/isbn/0-19-825029-0   (327 words)

Ebbinghaus, Flum, Thomas. Mathematical Logic. With respect to second-order logic (assuming an infinite language as above), the notions of fin-satisfiability and fin-validity of sentences are defined (in which we only look at finite structures). Lindstrom's Second Theorem says that any effective regular logical system effectively stronger than first-order logic which satisfies Lowenheim-Skolem and is enumerable for validity is actually effectively equivalent to first-order logic. First-order logic, as well as the extensions considered in Chapter 9, are examples of such a system. www.andrew.cmu.edu /user/cebrown/notes/ebbinghaus.html   (1936 words)

Second-Order Logic The formulas of second-order logic are those of first-order logic but variables may also vary over the predicate functors though it may be helpful to use different variables for individual constants and functors. In this formulation of second-order logic there are just the quantifiers used in first-order logic. An application of second-order logic includes individual constants, predicate constants, and formulas (non-logical axioms). cs.wwc.edu /~aabyan/Articles/GodelOntological/node5.html   (139 words)

Descriptive Complexity In 1974 Fagin gave a characterization of nondeterministic polynomial time as the set of properties expressible in second-order existential logic. Deterministic Logspace is equal to the set of boolean queries expressible in first-order logic, extended by a deterministic transitive closure operator. Nondeterministic Logspace is equal to the set of boolean queries expressible in first-order logic, extended by a transitive closure operator. www.cs.umass.edu /~immerman/descriptive_complexity.html   (990 words)

ON GÖDEL AND DBMS The second-order logic is of importance, however: as Gödel proved in his PhD dissertation in 1930, the first-order logic is both complete and consistent. It is, however, impossible to implement arithmetic in first order logic. Date's answer to the questions regarding Russel and Gödel, being a "logician trainee"  (when I'm not a DBA on some platforms which pretend to be relational, I'm a philosophy student) I have some comments. www.dbdebunk.com /page/page/1073010.htm   (432 words)

Existential second-order logic over graphs: charting the tractability frontier We study the complexity of evaluating existential second-order formulas that belong to prefix classes of existential second-order logic, where a prefix class is the collection of all existential second-order and the first-order quantifiers obey a certain quantifier pattern. We completely characterize the computation complexity of prefix classes of existential second-order logic in three different contexts: over directed graphs; over undirected graphs with self-loops; and over undirected graphs without self-loops. Citation:  G. Gottlob, P.G. Kolaitis, T. Schwentick, "Existential second-order logic over graphs: charting the tractability frontier," focs, p. csdl.computer.org /comp/proceedings/focs/2000/0850/00/08500664abs.htm   (278 words)

Reviews.com The classes of mu-calculus formulas that are characterized by the fragment of monadic second-order logic are themselves known to be as expressive as tree automata with Büchi conditions. The fragment of a second-order logic hinted at above is characterized to be bisimulation invariant; namely, if a tree is a model for such a formula, then any tree bisimilar to the first tree is also a model of the formula. The first section of the paper motivates the investigation, and states the main result the paper proposes to achieve, namely, that a certain fragment of monadic second-order logic is equivalent to certain classes of formulas of the mu-calculus. www.reviews.com /review/review_review.cfm?review_id=131977&listname=highlight   (630 words)

PHIL 411: Advanced Logic A first order logic sentence is consistent iff a certain argument in second-order logic (i.e., EInf /\ EG(f/D)) is valid. Functions are just 2-place relations whose extensions are sets of ordered pairs, where no two pairs have the same first member (input/argument) but different second members (output/value). P3:  If there is an effective test for the consistency of any sentence of first-order logic f, then there is an effective test for the validity of arguments of first-order logic. www.siue.edu /~wlarkin/teaching/PHIL411/undecidability.html   (704 words)

First-order Logic The semantics for first-order logic are usually Tarskian semantics. There are two ways to extend the language to use different sets of constants (a many-sorted first-order logic). An application of first-order logic includes individual constants, predicate constants, and formulas (non-logical axioms). cs.wwc.edu /~aabyan/Articles/GodelOntological/node4.html   (147 words)

Dissatisfaction with second-order logic Now back to my criticism of second-order logic: in light of the above, it seems that the theorem I quoted above about the "Dedekind completeness" of any real closed field ought to be a theorem in the second-order theory of real closed fields, because it's a statement about all first-order relations. For a while I've suspected what I consider to be a flaw in second-order logic (at least its interpretation), and it's recently been confirmed. From Mathematical Logic by H.-D. Ebbinghaus, J. Flum, and W. Thomas, we have part of the definition of "satisfaction" for a second-order assignment www.physicsforums.com /showthread.php?p=861635   (824 words)

03: Mathematical logic and foundations In second-order logic, the quantifiers are allowed to apply to relations and functions -- to subsets as well as elements of a set. Mathematical Logic is the study of the processes used in mathematical deduction. Just as ordinary logic may be studies with Boolean algebras one may formalize the calculi with many-valued logics using other algebraic systems. www.math.niu.edu /~rusin/known-math/index/03-XX.html   (2050 words)

Oxford University Press: Foundations without Foundationalism: Stewart Shapiro The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. www.us.oup.com /us/catalog/general/subject/Philosophy/LogicMathematics/~~/c2Y9YWxsJnNzPWF1dGhvci5hc2Mmc2Q9YXNjJnBmPTIwMCZ2aWV3PXVzYSZwcj0xMCZib29rQ292ZXJzPXllcyZjaT0wMTk4MjUwMjkw   (440 words)

Logicomp: Favorite Logic of October 2005: Monadic Second-Order Logic: Anthony Widjaja To's Blog on Logic and Complexity Monadic second-order logic (MSO) is a natural extension of first-order logic with quantifiers over sets of elements in the universe. Now, rather than babble about the current logic under consideration, I shall just sample some results that are particularly interesting from my point of view. These days, this result is widely employed in the area of "Foundations of XML", which have put automata, logic, and complexity theory under the same roof, which is no short of wondrous. logicomp.blogspot.com /2005/10/favorite-logic-of-october-2005-monadic.html   (731 words)

CSLI Calendar, 02 May 1996, vol.11:25 The first talk will consider how large a fragment of second-order logic is needed to represent the plural noun phrases and to which degree there are English sentences which distinguish between different possible consequence relations for second-order logic. I conjecture that they are recursively isomorphic to second order logic, but I currently have no proof of this. But second-order logic is not on the same firm grounds as first-order logic. www-csli.stanford.edu /Archive/calendar/1995-96/msg00025.html   (1724 words)

Citebase - On Spatial Conjunction as Second-Order Logic These embeddings show that the satisfiability of formulas in first-order logic with spatial conjunction is equivalent to the satisfiability of formulas in second-order logic. The embedding into spatial conjunction also has useful consequences: because a restricted form of spatial conjunction in two-variable logic preserves decidability, we obtain that a correspondingly restricted form of second-order quantification in two-variable logic is decidable. We construct an embedding from first-order logic with spatial conjunction into second-order logic, and more surprisingly, an embedding from full second order logic into first-order logic with spatial conjunction. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:cs/0410073   (669 words)

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